Problem 20 Solve each of the inequalities a... [FREE SOLUTION]

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Intermediate Algebra

Jerome E. Kaufmann, Karen L. Schwitters

$Math Studyset 91影视 Explanations$ Math

9 Edition

Chapter 2: Problem 20

Solve each of the inequalities and graph the solution set on a number line. $-3 x \leq-9$

Short Answer

Expert verified

The solution is $x \geq 3$.

Step by step solution

Isolate the Variable

The given inequality is $-3x \leq -9$. Our first task is to isolate $x$. To do this, we divide both sides of the inequality by $-3$. When we divide by a negative number, we need to flip the inequality sign. So, we get: \[ x \geq \frac{-9}{-3} \] Simplifying the fraction, we have: \[ x \geq 3 \]

Graph the Solution on a Number Line

Now that we have $x \geq 3$, we need to represent this inequality on a number line. To do this, we place a closed dot at $3$ on the number line because the inequality $x \geq 3$ includes $3$ (as indicated by the $\geq$ sign). We then shade the number line to the right of $3$ to indicate that $x$ can be any number greater than or equal to $3$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Number Line Graphing

Graphing solutions on a number line helps visually represent which numbers satisfy the inequality. For the inequality we solved, where the solution was $x \geq 3$, here's how it's done:

Find the number 3: On the number line, locate the number 3.
Closed dot for $\geq$: Place a closed dot (or circle) on the number 3. This indicates that the number 3 itself is included in the solution set.
Shade to the right: Since the inequality is $x \geq 3$, you shade all the numbers to the right of 3. This means any number greater than or equal to 3 is included in the solution.

Number line graphs are essential for showing solutions to inequalities as they provide a clear visual for both negative and positive solutions, helping you understand the range of numbers involved.

Variable Isolation

Variable isolation is the method used to solve for a variable, meaning you rearrange the equation or inequality to get the variable on one side by itself. In this exercise, we started with the inequality $-3x \leq -9$.

Here's a simple step-by-step approach:

Both sides of the inequality: Whatever you do to one side of the inequality, must be done to the other to maintain balance.
Divide by $-3$: To isolate $x$, divide both sides of the inequality by $-3$. Since this is a negative number, this action requires flipping the inequality sign from $\leq$ to $\geq$.
Resulting equation: This leaves us with $x \geq 3$, meaning $x$ is any number that is 3 or greater.

These steps highlight the importance of operating equally on both sides in equations and inequalities, especially when negative coefficients are involved.

Negative Coefficient Inequality

A negative coefficient in an inequality means the term multiplied by the variable is negative. In solving inequalities like $-3x \leq -9$, dividing or multiplying by a negative number requires extra care.

Here's what to keep in mind:

Flipping the sign: Dividing or multiplying both sides of an inequality by a negative flips the inequality sign. Here, dividing by $-3$ flips $\leq$ to $\geq$.
Balancing: Always ensure you perform the same operation on both sides. This ensures the inequality remains true and balanced.
Solution check: After solving, substitute numbers from your solution set back into the original inequality to ensure they satisfy it. For instance, substituting 3 back into $-3x\leq -9$ confirms the solution as valid.

Understanding this concept is pivotal in managing equations and inequalities effectively, ensuring solutions remain correct through each step.

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91影视

Solve each of the inequalities and graph the solution set on a number line. \(-3 x \leq-9\)

Short Answer

Step by step solution

Isolate the Variable

Graph the Solution on a Number Line

Key Concepts

Number Line Graphing

Variable Isolation

Negative Coefficient Inequality

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